Calculus: Differentiation, Integration and Applications
What you must command
Introductory calculus questions test rules and interpretation. For differentiation, know the power rule, constant and sum rules, and how to differentiate polynomials; understand gradient of a curve, tangents and normals, stationary points and their classification by the second derivative or by sign change; and apply calculus to rates of change and simple optimisation. For integration, know that it reverses differentiation, handle indefinite integrals with the constant of integration, and evaluate definite integrals as the difference of the antiderivative at the limits. Understand that a definite integral gives signed area under a curve, and that area between a curve and an axis may require splitting where the curve crosses the axis.
Traps that catch candidates
- Differentiate powers by reducing the index by one; integrate by raising it by one and dividing by the new index.
- The integral of 1/x is a logarithm, not a power of x.
- Always include the constant of integration for indefinite integrals.
- A stationary point is not automatically a maximum; classify it.
- For a normal, the gradient is the negative reciprocal of the tangent gradient.
- Definite integrals below the axis are negative, so split the interval when finding total area.
Worked example
Differentiate y = 3x^2 - 4x + 7. The derivative is 6x - 4. Setting it to zero gives x = 2/3. The second derivative is 6, which is positive, so this stationary point is a minimum.
Rule of thumb
Write the derivative as a gradient function and ask what the question wants: a value, a point, a nature, or an accumulated quantity. Naming the goal before calculating prevents most wasted work.